Fractional Hereditariness of Lipid Membranes: Instabilities and Linearized Evolution

TL;DR

This paper investigates the mechanical and viscoelastic behavior of lipid membranes, revealing how fractional hereditariness influences instabilities and long-tail relaxation dynamics through a generalized variational approach.

Contribution

It introduces a fractional hereditariness framework to model viscoelastic lipid membranes, extending classical elasticity with new bifurcation and relaxation insights.

Findings

Bifurcated modes with increased spatial oscillations due to fractional hereditariness.

Expanded range of areal stresses for material instabilities compared to elastic models.

Long-tail time decay in perturbation relaxation dynamics.

Abstract

In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elas- ticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is studied by minimizing the Gibbs free energy which penalizes perturbations of the changes of areal stretch and their gradients only [1]. As material instabilities arise whenever areal stretches characterizing homogeneous configurations lie inside the spinoidal zone of the free energy density, bifurcations from such configurations are shown to occur as oscillatory perturbations of the in-plane displacement. Experimental observations [2] show a power-law in-plane viscous behavior of lipid structures allowing for an effective viscoelastic behavior of lipid membranes [3], which falls in the framework of Fractional Hereditariness. A suitable generalization of the variational principle invoked for the elasticity is applied in this case, and the corresponding Euler-Lagrange equation is found together with a set of bound- ary and initial conditions. Separation of variables allows for showing how Fractional Hereditariness owes bifurcated modes with a larger number of spatial oscillations than the corresponding elastic analog. Indeed, the available range of areal stresses for material instabilities is found to increase with respect to the purely elastic case. Nevertheless, the time evolution of the perturbations solving the Euler-Lagrange equation above exhibits time-decay and the large number of spatial oscillation slowly relaxes, thereby keeping the features of a long-tail type time-response.